Proper Appeal

[Four Seam]

This thread relates to Irish Mafia's thread about a batter-runner failing to touch the orange part of a double base, but I'd like to ask a broader question about appeals.

The question is what constitutes a proper appeal?

Case 1. Base runner leaves third base before fly ball is caught in outfield. Defense plays ball to third base before runner can get back. Is it necessary for the third baseman to complete the appeal by saying to the BU she left too soon? Or does the mere act of playing the ball to third base and tagging the base complete the appeal so the runner is out? Am I right that the simple answer is no, she does not have to say anything to the BU?

Case 2. a. Base runner rounds third, missing the bag, and heads for home.
a.Ball is played to third base by defense and bag is touched before runner gets back to third. Is it necessary for third baseman to tell BU she missed third base to complete the appeal? Or does the mere act of playing the ball to that base and touching the base complete the appeal?

b.Runner gets back to third base and is standing on bag when third baseman catches ball and tags third base. Can she still be called out on appeal for missing the base? (Answer is obviously no, but I need to show this case for comparison to Case 3b below.)

Case 3. Batter-runner overruns first base after touching the white part of a double base instead of the orange part, ahead of the ball being caught by first baseman.
a. First baseman makes nice stretch and catches the ball with her foot touching first base before runner returns to first base. Is it necessary for first baseman to tell BU she missed the orange part of the bag, or does the mere act of touching the bag complete an appeal? Or is any appeal necessary...runner is out because she was in effect beaten by the throw because bag was touched before she touched orange part of base?

b. First baseman catches the ball, but before she touches first base or tags the runner, the runner gets back to first base and stops on the white part of the bag. First baseman then tags bag and appeals that runner missed the orange part of the bag. Is runner out, or does her having got back to the base before the appeal was made make her safe? Is this situation any different from Case 2b. above?

Thanks for replies from all blues who have patience to wade through this overly long thread.